arXiv:1912.12619v2 [math.CV] 15 Oct 2020

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SCHWARZIAN DERIVATIVES FOR PLURIHARMONIC

MAPPINGS

IASON EFRAIMIDIS, ALVARO FERRADA-SALAS, RODRIGO HERNANDEZ,AND RODRIGO VARGAS

Abstract. A pre-Schwarzian and a Schwarzian derivative for locally univalentpluriharmonic mappings in Cn are introduced. Basic properties such as thechain rule, multiplicative invariance and affine invariance are proved for theseoperators. It is shown that the pre-Schwarzian is stable only with respect torotations of the identity. A characterization is given for the case when thepre-Schwarzian derivative is holomorphic. Furthermore, it is shown that if theSchwarzian derivative of a pluriharmonic mapping vanishes then the analyticpart of this mapping is a Mobius transformation.

Some observations are made related to the dilatation of pluriharmonic map-pings and to the dilatation of their affine transformations, revealing differencesbetween the theories in the plane and in higher dimensions. An example is giventhat rules out the possibility for a shear construction theorem to hold in Cn, forn ≥ 2.

1. Preliminaries

1.1. Introduction. The pre-Schwarzian and Schwarzian derivatives of a locallyunivalent holomorphic function f in the plane are defined by

Pf = (log f ′)′ =f ′′

f ′and Sf = (Pf)′ − 1

2(Pf)2,

respectively. These operators appear frequently in geometric function theory andTeichmuller theory, where they are primarily used to prove criteria for univalenceas well as criteria for quasiconformal and homeomorphic extension. They satisfythe invariance relations

P (A f) = Pf and S(T f) = Sf

for linear (affine) maps A(z) = az+b, with a 6= 0, and for Mobius (linear fractional)transformations

T (z) =az + b

cz + d, ad− bc 6= 0.

2010 Mathematics Subject Classification. 30C99, 30G30, 31C10, 32A30, 32U05.Key words and phrases. Pluriharmonic mapping, pre-Schwarzian derivative, Schwarzian

derivative.1

http://arxiv.org/abs/1912.12619v2

2 I. EFRAIMIDIS, A. FERRADA-SALAS, R. HERNANDEZ, AND R. VARGAS

More generally, they satisfy the chain rules

P (g f) = (Pg f)f ′ + Pf and S(g f) = (Sg f)(f ′)2 + Sf

whenever the composition g f is well defined. We refer the reader to [15] and [19]for surveys of this classical theory.Both pre-Schwarzian and Schwarzian derivatives have been generalized and stud-

ied in the settings of harmonic mappings in the plane (see [3, 13]) and of holo-morphic mappings in C

n (see [21] and [11, 16, 18, 20], for example). The mainpurpose of this article is to extend both operators to pluriharmonic mappings inCn. We introduce a pre-Schwarzian derivative that simultaneously generalizes thepre-Schwarzian derivatives of the two more restricted settings given by Martın andthe third author [13] and Pfaltzgraff [21]. Moreover, we introduce a Schwarzianderivative that generalizes the Schwarzian operator for holomorphic mappings inCn defined by the third author [11] and based on the work of Oda [18].We will be using the notations Pf and Sf when we know that the mapping f is

holomorphic either in one or in several complex variables and the notations Pf andSf , with the mapping f as a subscript, in the more general settings of harmonicmappings in the plane or pluriharmonic mappings in Cn.A pluriharmonic mapping f in a simply connected domain Ω ⊂ Cn is a mapping

f = h + g, where h and g are holomorphic mappings in Ω with values in Cn. Weassume throughout the article that h is locally biholomorphic or, equivalently, thatdetDh(z) 6= 0 for all z ∈ Ω, where Dh is the n× n matrix

(

∂hi

∂zj

)

i,j=1,...n. Then the

dilatation of f is defined as ω = DgDh−1. The real Jacobian of f is given by

Jf (z) = det

(

Dh(z) Dg(z)

Dg(z) Dh(z)

)

, z ∈ Ω,

or equivalently, by

Jf(z) = |detDh(z)|2det(

In − ω(z)ω(z))

, z ∈ Ω, (1)

where In denotes the identity matrix of size n. Recent developments on plurihar-monic mappings include two-point distortion theorems [7, 9], a Schwarz Lemmaand Landau and Bloch theorems [2, 10], as well as theorems on stable univalentmappings and univalence criteria through the use of linearly connected domains [4],among others. A sufficient condition for f to be sense-preserving, i.e. for Jf > 0,was given in Theorem 5 of [7]. The norm notation hereafter refers to the standardoperator norm (see § 1.2).

Theorem A ([7]). If f = h + g is a pluriharmonic mapping in Ω for which‖ω(z)‖ < 1 for all z ∈ Ω then f is sense-preserving in Ω.

This was formulated in [7] for the case when Ω is the unit ball in Cn, but theproof does not depend on the domain of definition. In Section 4 we make the

SCHWARZIAN DERIVATIVES FOR PLURIHARMONIC MAPPINGS 3

simple observation that the converse of this fails, giving an example of a mappingf for which Jf > 0, but whose dilatation has arbitrarily large norm.Let A be a complex matrix that satisfies ‖A‖ < 1 and consider the affine mapping

of the form z 7→ z+Az. Post-composing with a pluriharmonic mapping f = h+ gwe obtain a new pluriharmonic mapping

F = f + Af = H +G

withH = h + Ag and G = g + Ah.

Assume now that ‖ω(z)‖ < 1 for all z ∈ Ω. Then clearly the mapping In +Aω(z)is invertible and we may compute the dilatation of F as

ωF =DGDH−1

=(

(ω + A)Dh) (

(In + Aω)Dh)−1

= (ω + A)(In + Aω)−1. (2)

Observe that in the plane, the above takes the familiar form of a disk-automorphismand its modulus is therefore bounded by 1. It is then natural to ask if the same holdsin dimension n ≥ 2, that is, if ‖ω‖ < 1 implies ‖ωF‖ < 1. However, the answer tothis is negative and therefore Theorem A can not be applied to F . Nevertheless,the assumption that ‖ω‖ < 1 is sufficient for F to be sense-preserving. We havethat

(i) ‖ω‖ < 1 does not imply ‖ωF‖ < 1

(ii) ‖ω‖ < 1 implies det(In − ωFωF ) > 0

(iii) det(In − ωω) > 0 does not imply det(In − ωFωF ) > 0.

Proposition (ii) is proved in Theorem 1, while counterexamples proving (i) and(iii) are given in Section 4, in Example 3 and Example 4, respectively. We perceivefrom this list that, in contrast to the planar case where we may apply an arbitrarynumber of affine transformations and each time obtain a sense-preserving harmonicmapping, it seems that in higher dimensions we are only allowed to apply one. Itis therefore interesting to ask the following.

Problem 1. Does there exist a condition on the dilatation ω which impliesdet(In − ωω) > 0 and det(In + Aω) 6= 0 for every matrix A with ‖A‖ < 1 and,moreover, is preserved under affine transformations?

In Section 2 we consider the class

P(Ω) = f : Ω → Cn | f pluriharmonic, Jh 6= 0 and Jf 6= 0 throughout Ω (3)

and define the pre-Schwarzian derivative of a mapping f = h + g in P(Ω) withdilatation ω as the bilinear mapping

Pf〈·, ·〉 = Ph〈·, ·〉 −Dh−1(

In − ωω)−1

ωDω〈·, Dh ·〉. (4)


We prove the chain rule for Pf and show that Pf is invariant under multiplicationby an invertible matrix and under affine transformations. We then further justifyour definition of Pf via best affine approximation. We characterize the case whenPf is holomorphic and deduce from it sufficient conditions for f to be univalent.We show that Pf is stable only with respect to rotations of the identity. We askthe following question for the unit ball Bn in Cn.

Problem 2. Does there exist a constant c ≤ 1 such that for every pluriharmonicmapping f in B

n the condition

(1− |z|2)‖Pf(z)‖ ≤ c, z ∈ Bn,

is sufficient for f to be univalent?

For f holomorphic Pfaltzgraff [21] answered this in the affirmative with c = 1,which is sharp.In Section 3 we define the Schwarzian derivative Sf of a pluriharmonic mapping f

and deduce its basic properties from the corresponding properties of Pf . We provethat if Sf vanishes then the holomorphic part of f is a Mobius transformation.

1.2. Preliminaries in several complex variables. Let Cn be the space of pointsz = (z1, . . . , zn), where each zi ∈ C. It is endowed with the inner (dot) productz · w =

∑ni=1 ziwi and the norm |z| = (z · z)1/2.

We denote by Lk(Cn) the space of continuous k-linear operators from Cn into

Cn. For T ∈ Lk(Cn) we write T 〈·, . . . , ·〉 to denote its placeholders. The standardoperator norm in Lk(Cn) is given by

‖T‖ = maxu1,...,uk∈Cn\0

|T 〈 u1

|u1|, . . . , uk

|uk|〉|.

When k = 1 we simply write L(Cn) for the space of linear maps and also write Tuinstead of T 〈u〉 for any linear map T .Let Ω be a domain in Cn and f a mapping in Ω with values in L(Cn). If

f is k-times (Frechet) differentiable with respect to z ∈ Ω then its k-th deriva-tive, denoted by Dkf(z), is a symmetric mapping in Lk+1(Cn), meaning that thevalue Dkf(z)〈u1, . . . , uk+1〉 remains unchanged after any permutation of the en-tries u1, . . . , uk+1 (see Theorem 14.6 in [17]). In the sequel we will be needing theproduct rule for the derivative of the product fg of two differentiable mappingsf, g : Ω → L(Cn), i.e., the composition of the linear mappings f(z) and g(z).Using the definition of differentiability one can easily prove that

D(fg)(z)〈·, ·〉 = Df(z)〈·, g(z) ·〉+ f(z)Dg(z)〈·, ·〉, z ∈ Ω. (5)

From this we can see that if f(z) is invertible for every z ∈ Ω then the derivativeof g(z) = f(z)−1 is given by

Dg(z)〈·, ·〉 = −g(z)Df(z)〈·, g(z) ·〉, z ∈ Ω. (6)


Finally, if f : Ω → L(Cn) is holomorphic then Taylor’s formula centered at someα ∈ Ω reads

f(z) =

∞∑

k=0

1

k!Dkf(α)(z − α)k, |z − α| < δ(α),

where δ(α) denotes the distance from α to the boundary of Ω; see Theorem 7.13in [17]. Here the notation Dkf(α)wk should be understood as Dkf(α)〈w, . . . , w, ·〉,with the point w repeated k times and one placeholder left without being evaluated.

1.3. Pluriharmonic mappings. A function u of class C2 defined in a domain Ωof Cn is called pluriharmonic if its restriction to every complex line is harmonic,that is, if for every fixed z ∈ Ω and direction θ ∈ C

n, |θ| = 1, the function u(z+ζθ)is harmonic in ζ ∈ C : z + ζθ ∈ Ω. Equivalently, u is pluriharmonic if

∂2u

∂zj∂zk= 0, for all j, k = 1, 2, . . . n.

Interest in these functions stems from the fact that in a simply connected domainthe class of real-valued pluriharmonic functions coincides with the class consistingof the real part of holomorphic functions. In contrast to the planar case, whereevery real-valued harmonic function is the real part of a holomorphic function,in dimension n ≥ 2 the pluriharmonic functions form a proper subclass of har-monic functions. See [14] or [22] for basic facts about pluriharmonic functions.A pluriharmonic mapping f : Ω → Cn is a mapping all of whose coordinates arecomplex-valued pluriharmonic functions.If Ω is simply connected then every pluriharmonic mapping f : Ω → Cn can be

written as f = h + g, with h and g holomorphic in Ω. To see this we may assumethat f has values in C by considering its coordinates. We write f = u+ iv. SinceΩ is simply connected we can analytically complete the pluriharmonic functionsu and v throughout Ω (see Theorem 4.4.9 in [22] and the comments thereafter),that is, we can find holomorphic mappings a and b in Ω for which u = Re a andv = Re b. Setting h = 1

2(a+ ib) and g = 1

2(a− ib) we easily verify that f = h+ g.

This representation is unique up to an additive constant. We say that it is thecanonical representation of f if g(z0) = 0 for some fixed point z0 in Ω.

1.4. Affine transformations. Let f = h + g be a pluriharmonic mapping withdilatation ω and let A be a complex matrix that satisfies ‖A‖ < 1. Then the affinetransformation F = f + Af has dilatation ωF = (ω + A)(In + Aω)−1, as seen in(2). The following theorem shows that the assumption ‖ω‖ < 1 is sufficient for Fto be sense-preserving. Its statement is purely about matrices though we maintainthe notation of pluriharmonic mappings in order to keep it in context.


Theorem 1. If the complex matrices A and ω satisfy ‖A‖ < 1 and ‖ω‖ < 1 thenthe matrix

ωF = (ω + A)(In + Aω)−1

satisfies

In − ωFωF = (In − AA)(In + Aω)−1(In − ωω)(In + Aω)−1

and det(In − ωFωF ) > 0.

Proof. Let B = In − ωFωF and compute

C = B(In + Aω)

= In + Aω − (ω + A)(In + Aω)−1(ω + A).

We add and subtract the term AA to get

C = In −AA +[

A− (ω + A)(In + Aω)−1]

(ω + A)

= In −AA +[

A(In + Aω)− (ω + A)]

(In + Aω)−1(ω + A)

= In −AA− (In − AA)ω (In + Aω)−1(ω + A).

Setting

D = (In − AA)−1C

= In − ω (In + Aω)−1(ω + A)

and noting thatω (In + Aω)−1 = (In + ωA)−1ω

we compute

D = In − (In + ωA)−1ω (ω + A)

= (In + ωA)−1[

In + ωA− ω (ω + A)]

= (In + ωA)−1(In − ωω).

We complete the factorization of B by writing

B = C (In + Aω)−1

= (In −AA)D (In + Aω)−1

= (In −AA)(In + ωA)−1(In − ωω)(In + Aω)−1.

In order to show that det(In−ωFωF ) > 0 we may replicate an idea from the proofof Theorem 5 in [7]: Note first that the assumptions ‖A‖ < 1 and ‖ω‖ < 1 and theabove factorization readily imply that det(In−ωFωF ) 6= 0. Consider parameters sand t in [0, 1] and see that all previous arguments apply to the matrices sA and tωso that the corresponding determinant is also different from zero. Since it is equalto 1 for s = t = 0, continuity with respect to s and t shows that it is positive fors = t = 1.


2. Pre-Schwarzian derivative

2.1. Previous definitions of the pre-Schwarzian. For a locally biholomorphicmapping f : Ω → Cn, where Ω is some domain in Cn, we adopt the definition ofthe pre-Schwarzian derivative as a bilinear mapping given by

Pf(z)〈·, ·〉 = Df(z)−1D2f(z)〈·, ·〉, z ∈ Ω. (7)

This was introduced by Pfaltzgraff in [21], who mostly considered the linear map-ping Pf(z)〈z, ·〉.On the other hand, for a locally univalent harmonic mapping f = h + g on a

planar domain Ω, with dilatation ω = g′/h′ : Ω → D, the definition

Pf(z) = (log Jf(z))z = Ph(z)− ω(z)ω′(z)

1− |ω(z)|2 , z ∈ Ω, (8)

was introduced in [13] by Martın and the third author. Here, in accordance with(1), the Jacobian is Jf = (1 − |ω|2)|h′|2. Note that the anti-analytic term in theexpression

log Jf = log(1− |ω|2) + log h′ + log h′

plays no role in the differentiation and, hence, we may use the expression

U = (1− |ω|2)h′ (9)

and obtain the exact same pre-Schwarzian by setting Pf = (logU)z. Exploitingthis observation we propose a definition of the pre-Schwarzian derivative for pluri-harmonic mappings in several complex variables.

2.2. Definition of Pf for pluriharmonic mappings. Let Ω be a simply con-nected domain in Cn and f ∈ P(Ω), the class of pluriharmonic mappings givenin (3). We consider

U(z) =(

In − ω(z)ω(z))

Dh(z), z ∈ Ω,

and define the pre-Schwarzian derivative of f as the mapping Pf : Ω → L2(Cn)given by

Pf(z)〈·, ·〉 = U(z)−1 DU(z)〈·, ·〉, z ∈ Ω. (10)

We apply the product rule (5) and, suppressing the variable z ∈ Ω, compute

DU〈·, ·〉 = −ωDω〈·, Dh ·〉+ (In − ωω)D2h〈·, ·〉.Therefore, we obtain from (10) that

Pf 〈·, ·〉 = Dh−1D2h〈·, ·〉 −Dh−1(

In − ωω)−1

ωDω〈·, Dh ·〉,which is equivalent to the expression (4). This operator is evidently a general-ization of both the pre-Schwarzian for holomorphic mappings in several complexvariables (7) and the pre-Schwarzian for planar harmonic mappings (8). Further


justification for this definition will be provided in § 2.4 with the process of bestaffine approximation.

2.3. Chain rule, multiplicative invariance and affine invariance. We firstestablish the chain rule for the composition of a pluriharmonic mapping with alocally biholomorphic mapping. Hereafter we tacitly assume in all statementsthat the necessary assumptions on the mappings involved hold so that their pre-Schwarzian derivatives are well defined.

Theorem 2 (Chain rule for Pf). Let f = h+g be a pluriharmonic mapping and ϕbe a locally biholomorphic mapping such that the composition f ϕ is well defined.Then

Pfϕ(z)〈·, ·〉 = Dϕ(z)−1Pf

(

ϕ(z))

〈Dϕ(z) ·, Dϕ(z) ·〉+ Pϕ(z)〈·, ·〉for all z in the domain of definition of ϕ.

Proof. We set F = f ϕ and write F = H + G, where H = h ϕ and G = g ϕ.We also note that ωF = ω ϕ. In order to use formula (4) for the mapping F wecompute

DH(z) = Dh(

ϕ(z))

Dϕ(z)

and

D2H(z)〈·, ·〉 = D2h(

ϕ(z))

〈Dϕ(z)· , Dϕ(z)·〉+Dh(

ϕ(z))

D2ϕ(z)〈·, ·〉.Therefore

PH(z)〈·, ·〉 =DH(z)−1D2H(z)〈·, ·〉=Dϕ(z)−1Ph

(

ϕ(z))

〈Dϕ(z) ·, Dϕ(z) ·〉+ Pϕ(z)〈·, ·〉.Finally, we compute

DωF (z)〈·, ·〉 = Dω(

ϕ(z))

〈Dϕ(z)· , ·〉and arrive at the desired conclusion after a substitution in (4).

The following is a generalization of Lemma 1 in [13]. It shows that the pre-Schwarzian of a pluriharmonic mapping is equal to the pre-Schwarzian of a specificholomorphic mapping which, however, depends on the point of evaluation.

Lemma 1. Let f = h + g be a pluriharmonic mapping in Ω having dilatation ω.Then

Pf(z0) = P(

h− ω(z0)g)

(z0)

for any fixed z0 in Ω.

Proof. Let F = h+ Ag, where A is a matrix with ‖A‖ < 1. We compute

DF = (In + Aω)Dh


and use the product rule (5) in order to find that

D2F 〈· , ·〉 = ADω〈· , Dh·〉+ (In + Aω)D2h〈· , ·〉.Therefore, in view of definition (7) we have that

PF 〈· , ·〉 = Ph〈· , ·〉+Dh−1(In + Aω)−1ADω〈· , Dh·〉.We now get the desired result by taking A = −ω(z0) and evaluating at z0.

Note that the pre-Schwarzian (7) of a holomorphic mapping f remains unchangedif f is multiplied by an invertible n× n complex matrix A. We readily see this bywriting F = Af and computing

PF = DF−1D2F = (ADf)−1AD2f = (Df)−1D2f = Pf. (11)

We now generalize this to pluriharmonic mappings.

Theorem 3 (Multiplicative invariance for Pf). If f is a pluriharmonic mappingand A is an invertible matrix then

PAf = Pf .

Proof. Let f = h + g and set F = Af = H +G, where H = Ah and G = Ag. Wecompute the dilatation of F as

ωF = DGDH−1 = (ADg)(ADh)−1 = AωA−1.

From Lemma 1 and equation (11) we get that

PF (z0) =P(

H − ωF (z0)G)

(z0)

=P[

A(

h− ω(z0)g)]

(z0)

=P(

h− ω(z0)g)

(z0)

=Pf(z0),

which completes the proof since z0 was arbitrary.

We now show that the pre-Schwarzian derivative (4) is invariant under compo-sition with affine transformations.

Theorem 4 (Affine invariance for Pf). If f is a pluriharmonic mapping then

Pf+Af = Pf

for every matrix A with ‖A‖ < 1.

Proof. We write f = h+g and set F = f+Af , for which we have the decompositionF = H +G, where H = h+Ag and G = g+Ah. According to (2), the dilatationof F is given by ωF = (ω + A)(In + Aω)−1. In order to solve for ω we write

(In − ωFA)ω = ωF − A.


We claim that In − ωFA is an invertible matrix. To see this we first note theelementary equality

(In + Aω)−1A = A(In + ωA)−1

and then compute

In − ωFA = In − (ω + A)(In + Aω)−1A

= In − (ω + A)A (In + ωA)−1

=[

In + ωA− (ω + A)A]

(In + ωA)−1

=(In − AA)(In + ωA)−1,

which proves our claim. Therefore we have

ω = (In − ωFA)−1(ωF − A ). (12)

We fix a point z0 and make the following computation:

H − ωF (z0)G =h+ Ag − ωF (z0)(g + Ah)

=(

In − ωF (z0)A)

h+(

A− ωF (z0))

g

=(

In − ωF (z0)A)

(

h +(

In − ωF (z0)A)−1(

A− ωF (z0))

g)

=(

In − ωF (z0)A)(

h− ω(z0)g)

,

where we made use of (12) at the last step. Since In − ωF (z0)A is a constant andinvertible matrix we have, in view of (11), that

P(

H − ωF (z0)G)

= P(

h− ω(z0)g)

.

Now, with two applications of Lemma 1 we get that

PF (z0) = P(

H − ωF (z0)G)

(z0) = P(

h− ω(z0)g)

(z0) = Pf (z0).

Since z0 was arbitrary the proof is complete.

2.4. Best affine approximation. Our definition is primarily justified by the factthat it coincides with the second (analytic) derivative of the affine deviation of f .To see this let T be the best affine approximation of f at the origin, i.e., the affinemap that agrees with f at its value and first analytic and anti-analytic derivatives.Clearly, we may assume that f(0) = 0 and, in view of Theorem 3, that Dh(0) = In.Hence

T (z) = z + ω(0) z.

The affine deviation of f is then defined by F = T−1 f . We compute F =

C(

f − ω(0) f)

, where C =(

In − ω(0)ω(0))−1

. Writing F = H +G we find that

H = C(

h− ω(0) g)

and G = C(

g − ω(0)h)

,


and, therefore, that

DH = C(

In − ω(0)ω)

Dh and DG = C(

ω − ω(0))

Dh.

Since DH(0) = In and DG(0) = 0 we get from formulas (4) and (7) that

PF (0) = PH(0) = D2H(0).

Noting that Pf = PF by the affine invariance shown in Theorem 4 we deduce thatPf(0) = D2H(0).For an arbitrary point a ∈ Ω we consider the translation fa(z) = f(z + a) and

argue as above in order to compute the analytic part of the affine deviation Fa offa at the origin as

Ha(z) = Dh(a)−1(

In − ω(a)ω(a))−1(

h(z + a)− ω(a) g(z + a))

,

where z ∈ Ω− a. Hence, we find that

Pf(a) = Pfa(0) = PFa(0) = PHa(0) = D2Ha(0)

and conclude that our definition of the pre-Schwarzian derivative is equal to thesecond (analytic) derivative of the affine deviation of a pluriharmonic mapping and,thus, being in line with ideas of Cartan [1] on the Schwarzian derivative, furtherexploited by Tamanoi [23], it constitutes a natural definition.

2.5. Holomorphic or vanishing pre-Schwarzian. For a holomorphic map f itis elementary to see from definition (7) that if Pf = 0 in some open set then f islinear, that is, f(z) = Az + b, for some invertible matrix A and some b ∈ Cn.In the plane on the other hand, in view of definition (8), if the pre-Schwarzian of

a harmonic mapping f = h + g is holomorphic then a differentiation with respectto z gives

ω′ω′

(1− |ω|2)2 = 0,

from which it follows that ω is constant.However, it is relatively easy to produce examples of pluriharmonic mappings in

C2 with vanishing pre-Schwarzian (4) but non-constant dilatation.

Example 1. Let φ be a holomorphic function in some domain Ω1 in C and considerthe mapping

f(z, w) =(

z, w + φ(z))

which is pluriharmonic in Ω1 ×C and also, clearly, univalent. A representation forf is given by

h(z, w) = (z, w) and g(z, w) =(

0, φ(z))

,

and its dilatation is

ω(z, w) = Dg(z, w) =

(

0 0φ′(z) 0

)

,


which is non-constant if we choose φ to be non-linear. For the pre-Schwarzian off we easily compute

U =(

I2 − ω ω)

Dh = I2

and therefore Pf = 0 in view of (10).

Our main theorem here characterizes the case when the pre-Schwarzian deriva-tive of a pluriharmonic mapping in the general class (3) is holomorphic.

Theorem 5. Let f ∈ P(Ω) be such that ω(z0) = 0 for some z0 ∈ Ω. Then Pf isholomorphic if and only if ωω ≡ 0.

Proof. Observe first that if ωω ≡ 0 then a differentiation (after a conjugation)shows that ωDω ≡ 0. In view of (4) we have that Pf = Ph which is clearlyholomorphic.Conversely, if Pf is holomorphic then, by formula (4), we have that

0 = DPf 〈·, ·, ·〉 = −Dh−1D[

(In − ωω)−1 ω] ⟨

·, Dω〈·, Dh ·〉⟩

.

Since Dh is invertible this is equivalent to

D[

(In − ωω)−1 ω] ⟨

·, Dω〈·, ·〉⟩

= 0. (13)

Using the differentiation properties (5) and (6) we compute

D[

(In − ωω)−1 ω]

〈·, ·〉 = (In − ωω)−1Dω〈·, ω(In − ωω)−1ω ·〉+ (In − ωω)−1Dω〈·, ·〉= (In − ωω)−1Dω〈·,

(

ωω(In − ωω)−1 + In)

·〉= (In − ωω)−1Dω〈·, (In − ωω)−1 ·〉,

so that (13) is equivalent to

Dω⟨

·, (In − ωω)−1Dω〈·, ·〉⟩

= 0. (14)

In order to differentiate (14) with respect to z we compute

D[

(In − ωω)−1Dω]

〈·, ·, ·〉 =(In − ωω)−1Dω⟨

·, ω(In − ωω)−1Dω〈·, ·〉⟩

+ (In − ωω)−1D2ω〈·, ·, ·〉

and apply to this the operator Dω. Then a further usage of (14) shows that thefirst summand vanishes so that we obtain

Dω⟨

·, (In − ωω)−1D2ω〈·, ·, ·〉⟩

= 0.

This can be repeated an arbitrary number of times, so that after k− 1 differentia-tions of (14) we have that

Dω⟨

·, (In − ωω)−1Dkω〈·, . . . , ·〉⟩

= 0, k ≥ 1, (15)


holds throughout the domain Ω. According to the Taylor formula centered at someα in Ω we have that

ω(z) =∞∑

k=0

1

k!Dkω(α)(z − α)k, |z − α| < δ(α), (16)

where δ(α) = dist(α, ∂Ω). We apply to this the operator Dω〈·, (In − ωω)−1·〉evaluated at α so that, in view of (15), we obtain

Dω(α)⟨

·,(

In − ω(α)ω(α))−1

ω(z) ·⟩

= Dω(α)⟨

·,(

In − ω(α)ω(α))−1

ω(α) ·⟩

= Dω(α)⟨

·, ω(α)(

In − ω(α)ω(α))−1 ·

⟩

.

Since α ∈ Ω is arbitrary we may take z = z0 in order to use the assumptionω(z0) = 0. Denoting by

E(z) = α ∈ Ω : |α− z| < δ(α)the set of points in Ω which lie closer to z than to ∂Ω (that is, ∂E(z) consists ofpoints which are equidistant to z and ∂Ω), we get that

Dω(α)⟨

·, ω(α)(

In − ω(α)ω(α))−1 ·

⟩

= 0, α ∈ E(z0).

Equivalently, we have that Dω〈·, ω ·〉 = 0 in E(z0). After k − 1 differentiations weget that

Dkω〈·, . . . , ·, ω ·〉 = 0, k ≥ 1,

in E(z0). We consider again the Taylor formula (16) centered at some α ∈ E(z0)

and multiply it on the right with ω(α) in order to obtain

ω(z)ω(α) = ω(α)ω(α), |z − α| < δ(α).

Taking z = z0 we conclude that ωω = 0 in E(z0). In view of the identity prin-ciple for real analytic functions (see Corollary 2.3.8 in [14]) we have that ωω = 0throughout Ω.

Note that if f ∈ P(Ω) does not satisfy the second assumption of Theorem 5

then we may normalize it by means of the affine transformation F = f − ω(z0) f ,so that ωF (z0) = 0, while at the same time having PF = Pf in view of Theorem 4.Since the point z0 ∈ Ω is arbitrarily chosen it is easy to see that Pf is holomorphicif and only if

(

ω(α)− ω(β))(

In − ω(β)ω(α))−1(

ω(α)− ω(β))

= 0, for all α, β ∈ Ω.

As a direct consequence of Theorem 5, the case when Pf vanishes is characterizedin the following proposition.

Corollary 1. Let f ∈ P(Ω) be such that ω(z0) = 0 for some z0 ∈ Ω. ThenPf ≡ 0 if and only if ωω ≡ 0 and h is linear.


In order to see if any condition on Pf can imply that f is univalent we recallnow Corollary 2.2 from [4]. We formulate it here for a general simply connecteddomain Ω ⊂ C

n (since its proof does not depend on the domain of definition), eventhough in [4] it was stated for the unit ball in Cn.

Theorem B ([4]). Let h be a biholomorphic mapping in Ω for which h(Ω) is convexand let f = h+ g be a pluriharmonic mapping for which ‖ω‖ < 1 in Ω. Then f isunivalent.

With the aid of this we can prove the following.

Corollary 2. Let Ω be a convex domain and f ∈ P(Ω) be such that ‖ω‖ < 1 inΩ and ω(z0) = 0 for some z0 ∈ Ω. Then Pf ≡ 0 implies that f is univalent.

Proof. Since h is linear by Corollary 1, its image is a convex domain and Theorem Bmay be applied.

2.6. Stability for the pre-Schwarzian. Here we turn to the notion of stabilityas it was introduced in the plane in [12] and generalized to the following form in [4].A property is said to be stable if it is shared by f = h+ g and F = h+Ag for allunitary matrices A. Similarly, we say that an expression is stable if it is invariantunder the transformation f 7→ h+Ag. We show that the pre-Schwarzian is stableunder rotations of the identity and that these are the only unitary matrices withthis property.We denote the unit circle by T = z ∈ C : |z| = 1. Also, we denote by ej the

vectors of the standard basis and by Eij the matrix whose only non-zero entry isat the (i, j)-position and is equal to 1. We indicate with Rk on the left and Ck

on the top of a matrix the position of the k-th row and column, respectively. Forexample, the matrix Eij could be given by

Eij =

(

Cj

Ri 1)

,

where all entries that do not appear are equal to zero.In the following the standard notation f = h + g and F = h+ Ag is used.

Theorem 6. Let A be a unitary matrix. Then PF = Pf for all f ∈ P(Ω) with‖ω‖ < 1 if and only if A = λIn for some λ ∈ T.

Proof. The dilatation of F is given by ωF = Aω. Hence, in view of definition (4),it is evident that PF = Pf if A is a rotation of the identity.For the converse we give counterexamples in the polydisk Ω = Dn. Assume first

that the matrix A = (aij)ij has at least one non-zero entry off the diagonal, that is,aij 6= 0 for some i 6= j. We consider the mapping f = h+ g for which h(z) = z andg(z) = 1

2z2i ej . We see that ω = ziEji satisfies ‖ω‖ = |zi| < 1 and, moreover, that


ωω = 0, so that Pf = 0. The dilatation of the transformed mapping F is givenby ωF = Aω = ziM , where M is the matrix with non-zero elements only in thei-th column Ci = (a1j , . . . , anj). Since ωF (0) = 0 and ωFωF = aijziωF is not zero,Corollary 1 can be applied to deduce that PF is not zero and, therefore, distinctfrom Pf .For the remaining case we may assume that A is a diagonal matrix with entries

λk ∈ T, k = 1, 2, . . . , n, and such that λi 6= λj for some i < j. We first prove that,in general, PF = Pf is equivalent to

(AωA− ω)(In − ωω)−1Dω = 0. (17)

Indeed, in view of (4), PF = Pf is equivalent to

(

In − AωAω)−1

AωADω =(

In − ωω)−1

ωDω,

which is the same as

0 =[

AωA−(

In −AωAω)(

In − ωω)−1

ω]

Dω

=[

AωA−(

In −AωAω)

ω(

In − ωω)−1]

Dω

=[

AωA(

In − ωω)

−(

In − AωAω)

ω](

In − ωω)−1

Dω

= (AωA− ω)(In − ωω)−1Dω,

which proves our claim (17). We consider the mapping f = h + g with h(z) = zand g(z) = 1

2(z2i ej + z2j ei). We compute

ω =

Ci Cj

Ri zj

Rj zi

and Dω〈u, v〉 =

Ci Cj

Ri uj

Rj ui

v.


Moreover, we find that ωω = zizjEii+zizjEjj, so that In−ωω is a diagonal matrixwhich we can easily invert to obtain

(In − ωω)−1 =

Ci Cj

1. . .

1Ri

11−zizj

1. . .

1Rj

11−zizj

1. . .

1

.

Finally, we have

AωA− ω =

Ci Cj

Ri (λiλj − 1)zj

Rj (λiλj − 1)zi

,

with which we conclude that

(AωA− ω)(In − ωω)−1Dω〈u, v〉 =

Ci Cj

Ri

(λiλj−1)uizj1−zizj

Rj

(λiλj−1)ujzi1−zizj

,

Since this is not zero we get that Pf and PF are distinct in view of (17).

3. Schwarzian derivative

3.1. The Schwarzian derivative for holomorphic mappings in Cn. For alocally biholomorphic mapping f in a domain Ω ⊂ Cn, Oda [18] defined theSchwarzian derivatives

Skijf =

n∑

ℓ=1

∂2fℓ∂zi∂zj

∂zk∂fℓ

− 1

n+ 1

(

δik∂

∂zj+ δjk

∂

∂zi

)

log Jf , (18)

for i, j, k = 1, . . . , n; here δij is Kronecker’s delta. These differential operatorssatisfy a certain chain rule formula and, also, in dimension n ≥ 2 they all vanish


only for Mobius transformations, i.e., for the mappings

T (z) =

(

ℓ1(z)

ℓ0(z), . . . ,

ℓn(z)

ℓ0(z)

)

, z ∈ Cn, (19)

where ℓi(z) = ai0 + ai1z1 + . . .+ ainzn, i = 0, . . . , n, with det(aij)ij 6= 0. Note that,in contrast to the planar case, these are differential operators of order 2. This isexplained by the fact that in the plane the order of the Mobius group is 3 and,therefore, any Mobius-invariant operator should have order at least 3. However,in dimension n ≥ 2 the parameters involved in the value, the first and the secondderivatives of f are n2(n+1)/2+n2+n, already exceeding the order of the Mobiusgroup which is n2 + 2n. The difference of the two orders is n(n − 1)(n + 2)/2and this is precisely the number of independent terms Sk

ijf , which can be counted

considering that they satisfy Skijf = Sk

jif for all k and∑n

j=1 Sjijf = 0.

With the matrices Skf =(

Skijf)

ij, for k = 1, . . . , n, the third author [11] defined

the symmetric bilinear operator

Sf(z)〈u, v〉 =(

utS1f(z)v, . . . , ut

Snf(z)v

)

, z ∈ Ω, u, v ∈ Cn; (20)

here ut denotes the transpose of the vector u. A straightforward calculation thenproduces the expression

Sf(z)〈u, v〉 = Pf(z)〈u, v〉 − 1

n+ 1

(

(

∇ log Jf(z) · u)

v +(

∇ log Jf(z) · v)

u)

, (21)

where Pf is the pre-Schwarzian derivate (7) and ∇ = (∂/∂z1, . . . , ∂/∂zn) is thecomplex gradient operator. A way to verify that (20) and (21) are equivalent is toconsider the vectors u = ei and v = ej of the standard basis and see that in thiscase both expressions are equal to

(

S1ijf(z), . . . , S

nijf(z)

)

.

3.2. Definition of Sf for pluriharmonic mappings. For f = h+g in P(Ω), theclass of pluriharmonic mappings given in (3), we define the Schwarzian derivativeSf in complete analogy to (21), employing the pre-Schwarzian derivate (4) andthe Jacobian (1) of f . We now find two equivalent formulations of this definition.With a brief calculation (and suppressing the variable z ∈ Ω) we get that

Sf 〈u, v〉 = Sh〈u, v〉 −Dh−1(In − ωω)−1 ωDω〈u,Dh v〉 (22)

− 1

n+ 1

(

∇ log[

det(In − ωω)]

· u

v +

∇ log[

det(In − ωω)]

· v

u

)

.

Also, using Jacobi’s formula for the derivative of a determinant (see [8]), we getfrom (1) that

∂

∂zk(log Jf) = Tr

(

Dh−1∂(Dh)

∂zk

)

− Tr

(

(In − ωω)−1 ∂ω

∂zkω

)

,


where Tr(·) denotes trace. Observe that the linear mappingsD2h〈ek, ·〉 andDω〈ek, ·〉can be interpreted as the matrices ∂(Dh)/∂zk and ∂ω/∂zk, respectively. Usingstandard properties of the trace we get

∂

∂zk(log Jf) =Tr

(

Dh−1D2h〈ek, ·〉)

− Tr(

ω(In − ωω)−1Dω〈ek, ·〉)

=Tr (Ph〈ek, ·〉)− Tr(

Dh−1(In − ωω)−1ωDω〈ek, Dh·〉)

=Tr (Pf〈ek, ·〉) .Therefore, once again from (21), we obtain

Sf〈u, v〉 = Pf〈u, v〉 (23)

− 1

n + 1

(

[

n∑

k=1

Tr(

Pf〈ek, ·〉)

ek

]

· u

v +

[

n∑

k=1

Tr(

Pf 〈ek, ·〉)

ek

]

· v

u

)

,

which shows that Sf can be written only in terms of Pf .

3.3. Basic theorems. In view of formula (23) the following theorem is a directconsequence of the corresponding theorems on the pre-Schwarzian.

Theorem 7. Let f = h+ g be a pluriharmonic mapping in Ω having dilatation ω.Then

(i) Sf+Af = Sf for every matrix A with ‖A‖ < 1,

(ii) SBf = Sf for every invertible matrix B,

(iii) Sh+λg = Sf for every complex number λ with |λ| = 1,

(iv) Sf (z0) = S(h− ω(z0)g)(z0) for any fixed z0 ∈ Ω.

Proof. Statements (i), (ii), (iii) and (iv) follow directly from Theorem 4, Theo-rem 3, Theorem 6 and Lemma 1, respectively.

Theorem 8 (Chain rule for Sf). Let f = h+g be a pluriharmonic mapping and ϕbe a locally biholomorphic mapping such that the composition f ϕ is well defined.Then

Sfϕ(z)〈·, ·〉 = Dϕ(z)−1Sf

(

ϕ(z))

〈Dϕ(z) ·, Dϕ(z) ·〉+ Sϕ(z)〈·, ·〉for all z in the domain of definition of ϕ.

Proof. In view of the chain rule for the pre-Schwarzian given in Theorem 2 and theformula

Jfφ(z) = Jf

(

φ(z))

Jφ(z)


satisfied by the Jacobian of a composition, our definition (21) readily yields

Sfφ(z)〈u, v〉 = Sφ(z)〈u, v〉+Dϕ(z)−1Pf

(

ϕ(z))

〈Dϕ(z)u,Dϕ(z)v〉

− 1

n + 1

(

(

∇(log Jf φ)(z) · u)

v +(

∇(log Jf φ)(z) · v)

u)

.

To get the desired result it suffices to verify that

∇(log Jf φ)(z) · u = ∇(log Jf)(

φ(z))

·(

Dφ(z)u)

for the vectors u = ek of the standard basis or, equivalently, that

∂ (Jf φ)∂zk

(z) =

n∑

j=1

∂Jf

∂zj(φ(z))

∂φj

∂zk(z),

which is evident.

It is easy to see in (22) that if the dilatation ω is constant then Sf = Sh and,furthermore, that if f = T + AT for a Mobius transformation T as in (19) and amatrix A with ‖A‖ < 1 then Sf = ST = 0. In the converse direction we have thefollowing theorem.

Theorem 9. Let f = h + g ∈ P(Ω) be such that ω(z0) = 0 for some z0 ∈ Ω, andlet also ϕ be a holomorphic mapping in Ω. Then Sf = ϕ implies that Sh = ϕ.

Proof. Observe that

∂

∂zilog[

det(

In − ωω)]

=1

det(

In − ωω)

n∑

k=1

det(

F1, . . . ,∂Fk

∂zi, . . . , Fn

)

,

where Fk’s are the rows of In − ωω. But Fk = ek − ωkω, where ωk is the k-row ofω, thus

∂

∂zilog[

det(

In−ωω)]

=1

det(

In − ωω)

n∑

k=1

det(

e1−ω1ω, . . . ,−∂ωk

∂ziω, . . . , en−ωnω

)

.

Hence this expression vanishes at z0 since ω(z0) = 0. Moreover, differentiationwith respect to zj , 1 ≤ j ≤ n, of each row in the determinant in the summationabove will leave the term ω intact, that is,

∂

∂zj

(

−∂ωk

∂ziω

)

= − ∂2ωk

∂zj∂ziω and

∂

∂zj(eℓ − ωℓω) = −∂ωℓ

∂zjω, ℓ 6= k.

Therefore, we have that

∂mdet(

e1 − ω1ω, . . . ,−∂ωk

∂ziω, . . . , en − ωnω

)

∂zm1

1 · · ·∂zmnn

(z0) = 0, m ≥ 0, (24)

where m = m1 + . . .+mn, again because ω(z0) = 0.


On the other hand, we can rewrite Dn(

Dh−1(In − ωω)−1ωDωDh)

as

n∑

k=0

(

n

k

)

Dn−k(

Dh−1(In − ωω)−1)

ωDk(DωDh).

Hence,

Dn(

Dh−1(In − ωω)−1ωDωDh)

(z0) = 0. (25)

Now, using the equations (22), (24), and (25) we have that Dnϕ(z0) = DnSf (z0) =DnSh(z0) for all n ≥ 0 and, therefore, Sh = ϕ.

As a direct consequence of Theorem 9 for ϕ ≡ 0, and in view of the fact thatthe Schwarzian derivative (20) vanishes only for Mobius transformations (19), weget the following proposition.

Corollary 3. Let f = h + g ∈ P(Ω) be such that ω(z0) = 0 for some z0 ∈ Ω.Then Sf ≡ 0 implies that h is a Mobius transformation.

4. Miscellaneous

In this section we give examples which reveal some interesting differences be-tween harmonic mappings in the plane and pluriharmonic mappings in highercomplex dimensions.

4.1. Dilatation of sense-preserving pluriharmonic mappings. According toTheorem A, a pluriharmonic mapping f = h+ g is sense-preserving at a point z ifits dilatation satisfies ‖ω(z)‖ < 1. It is natural to ask if, conversely, the conditionJf > 0 can imply some bound on the norm of the dilatation. We now see that thisis not possible.

Example 2. Let h be a locally biholomorphic mapping in a domain in C2 andconsider the constant dilatation

ω =

(

0 t−1 0

)

, t ≥ 1.

We readily compute I2 − ωω = (1 + t)I2, hence Jf = |detDh|2(1 + t)2 > 0. But‖ω‖ = t, which can be arbitrarily large. The pluriharmonic mapping f = h+ g forwhich g = (g1, g2) = (th2,−h1) satisfies the above since, indeed, Dg = ωDh.It is easy to generalize this example to any dimension n ≥ 2. Simply take ω to

be the n × n matrix with only two non-zero entries: t in the upper-right cornerand −1 in the lower-left corner.


4.2. Dilatation of affine transformations. Let f = h + g be a pluriharmonicmapping and let A be a matrix with ‖A‖ < 1. Then, in view of (2), the affinetransformation F = f + Af has dilatation ωF = (ω + A)(In + Aω)−1. We nowgive a counterexample proving that ‖ω‖ < 1 does not imply ‖ωF‖ < 1. Note thatthe choice of A in this example is arguably the most useful since it ensures thatthe mapping F satisfies the normalization ωF (0) = 0.

Example 3. Let ϕ be an automorphism of the unit disk D = z ∈ C : |z| < 1given by ϕ(z) = (α + z)/(1 + αz), for α ∈ (0, 1), and consider the matrix valuedmapping

ω(z, w) =ϕ(z)√

2

(

1 10 0

)

, (z, w) ∈ D× Ω2,

for some domain Ω2 ⊂ C containing the origin. We readily compute that ‖ω(z, w)‖ =|ϕ(z)| < 1. Consider also the matrix

A = −ω(0, 0) = − α√2

(

1 10 0

)

and the affine transformation F = f + Af . We compute

ω + A =ϕ− α√

2

(

1 10 0

)

and

I2 + Aω =1

2

(

2− αϕ −αϕ0 2

)

,

whose inverse is given by

(I2 + Aω)−1 =1

2− αϕ

(

2 αϕ0 2− αϕ

)

.

Therefore, we have that

ωF =

√2 (ϕ− α)

2− αϕ

(

1 10 0

)

,

whose norm is

‖ωF (z, w)‖ =2 |ϕ(z)− α||2− αϕ(z)| =

2(1− α2)|z||2− α2 + αz| .

We compute

‖ωF‖∞ = sup‖ωF (z, w)‖ : (z, w) ∈ D× Ω2 =2(α + 1)

α + 2

and see that this increases with α ∈ (0, 1) from 1 to 4/3.


An obvious modification of this example to any dimension n ≥ 2 would be toconsider the dilatation

ω(z) =ϕ(z1)√

nB, z1 ∈ D,

with the matrix B having all the entries of its first row equal to 1 and all theremaining entries equal to 0. Then, with the same choice of ϕ and A, it is notdifficult to compute that ‖ωF‖∞ = n(α+1)/(α+n), which increases with α ∈ (0, 1)from 1 to 2n/(n+ 1).

The following example shows that under only the assumption det(In − ωω) > 0it is possible that the mapping ωF is not even well defined.

Example 4. Let t ∈ (0, 1) and consider the matrices

A =

(

t 00 −t

)

and ω =

(

0 1/t2

−1 0

)

,

interpreting ω as the dilatation of the pluriharmonic mapping

f(z, w) =

(

z +w

t2, w − z

)

,

for example. We easily compute that ‖A‖ = t and

In − ωω =

(

1 + 1/t2 00 1 + 1/t2

)

,

so that det(In − ωω) = (1 + 1/t2)2 > 0. Since the matrix

I2 + Aω =

(

1 1/tt 1

)

is singular we deduce that the holomorphic part H of the affine transformationF = f + Af is everywhere singular, that is, it satisfies detDH ≡ 0, and thereforeit is not possible to define the dilatation of F .

We note that the proof of Theorem 1 shows that the condition det(In−ωω) > 0together with the condition det(In + Aω) 6= 0 for every matrix A with ‖A‖ < 1are sufficient for det(In − ωFωF ) > 0. However, this does not give an answerto Problem 1 since the latter of these conditions is not preserved under affinetransformations.

4.3. No shear construction in Cn. A simply connected domain in the plane iscalled convex in the horizontal direction (CHD) if its intersection with any horizon-tal line is connected or empty. Let f = h+g be a locally univalent planar harmonicmapping (the domain of definition is not relevant here). Then according to Clunieand Sheil-Small’s “shear construction” introduced in [5] (see also [6, §3.4]), f isunivalent and its range is CHD if and only if h − g has the same properties. Thekey to proving this theorem is the following lemma.


Lemma C ([5]). Let a domain Ω be CHD and let p be a real-valued continuousfunction on Ω. Then the mapping w 7→ w + p(w) is univalent in Ω if and only ifit is locally univalent. If it is univalent then its range is CHD.

To generalize the shear construction in several complex variables we would havefirst to give a suitable analogue of the concept of directional convexity for a domainΩ in Cn and then to generalize Lemma C to any continuous p : Ω → Rn. However,we now give an example of a convex domain in C2 and a function p for whichw 7→ w+ p(w) is locally univalent but not univalent, thus excluding the possibilityof such a generalization.

Example 5. Consider the domain Ω = Ω1 × Ω2 ⊂ C2, where

Ω1 = w ∈ C : −1 < Rew < 1and

Ω2 = w ∈ C : −(π + ε) < Rew < π + ε,for some ε > 0. With the notation w = (w1, w2) and wk = xk + iyk, k = 1, 2, let

p(w) = ex1(cos x2, sin x2)− (x1, x2).

Setting q(w) = w + p(w) we may interpret q as a mapping in R4 by writing

q(x1, y1, x2, y2) = (ex1 cos x2, y1, ex1 sin x2, y2),

and compute its Jacobian as Jq(w) = e2x1 > 0. Therefore the mapping q is locallyunivalent. But, we note that

q(0,−π) = (−1, 0) = q(0, π),

which shows that q is not univalent in Ω.A simple modification of the function p, for example, by setting zero in the

remaining entries, serves as a counterexample in any dimension n ≥ 2.

Acknowledgements. We wish to thank professor Martin Chuaqui for many fruitfuldiscussions and, in particular, for an observation which significantly reduced thelength of the proof of Theorem 5.

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Department of Mathematics and Statistics, Texas Tech University, Box 41042,

Lubbock, TX 79409, United States

Email address : [email protected]

Facultad de Matematicas, Pontificia Universidad Catolica de Chile, Santiago.


Facultad de Ingenierıa y Ciencias, Universidad Adolfo Ibanez, Av. Padre Hur-

tado 750, Vina del Mar, Chile.


Facultad de Matematicas, Pontificia Universidad Catolica de Chile, Santiago.


arXiv:1912.12619v2 [math.CV] 15 Oct 2020

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